Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Write an inequality that represents the interval. Then state whether the interval is bounded or unbounded. \([-5, \infty)\)

Step-by-Step Solution

Answer

The inequality that represents the given interval \([-5, \infty)\) is \(x \geq -5\) and the interval is unbounded.

Step 1: Converting the interval into an inequality

An interval notation represents a range of values or a 'from ... to ...' relation. Here, \([-5, \infty)\) means 'from -5 up to infinity', including -5 (because of the bracket, not the parenthesis). To convert it into an inequality, we keep in mind that every value in the interval satisfies it. This means that every value \(x\) in the interval is greater than or equal to -5. So the inequality will be \(x \geq -5\)

Step 2: Determining if the interval is bounded or unbounded

An interval is said to be bounded if it has both upper and lower limits. Its unbounded if it only has either a lower limit or an upper limit. Here, \([-5, \infty)\) has a lower limit of -5. But it doesn't have an upper limit because \(\infty\) represents an infinity or endless value. Therefore, this interval is unbounded

Key Concepts

Interval Notation

Interval notation is a shorthand used in mathematics to describe a set of numbers within a specific range. Unlike inequalities, which explicitly state conditions, interval notation uses familiar mathematical symbols like brackets and parentheses to convey the range.

• Brackets \( [\) or \( ] \) indicate that the endpoint is included in the set, also known as a closed interval.
• Parentheses \( (\) or \( ) \) suggest that the endpoint is not included, known as an open interval.

For example, the interval \([-5, \infty)\) uses a bracket at \(-5\), showing that \(-5\) is part of the interval. The parenthesis at \(\infty\) indicates infinity itself isn't part of the interval. This notation efficiently communicates a range of numbers from \(-5\) to infinity, inclusive of \(-5\). Such concise expression is particularly useful in calculus and algebra for illustrating where certain conditions apply.

Bounded and Unbounded Intervals

Understanding whether an interval is bounded or unbounded is crucial in grasping the extent of the set of numbers it encompasses. Intervals refer to sections of the real number line, but these sections can either have finite limits—bounded—or extend indefinitely—unbounded.

• A bounded interval has both an upper and a lower limit. It contains all numbers within these bounds, not going beyond.
• An unbounded interval extends indefinitely in one or both directions, possessing either just an upper or just a lower limit, or none at all.

When we look at the interval \([-5, \infty)\), it's clear that it has only a lower bound at \(-5\), with no upper boundary. For this reason, \( \infty \), symbolizing an endless continuation, means the interval is unbounded in the upper direction.

Real Number Line

The real number line is an indispensable tool in mathematics for visualizing sets of numbers. It helps illustrate where intervals and their boundaries lie relative to other values.Conceptually, think of the real number line as an endless horizontal line where every point corresponds to a real number. It's essential in depicting how numbers, and intervals, are arranged and in what directions they extend.

• Points on the line represent specific numbers.
• Rays or segments along the line highlight intervals, stretching between or beyond certain values.

For instance, the interval \([-5, \infty)\) on the real number line would start at \(-5\) and extend rightward indefinitely. This provides a clear, visual method to understand abstract concepts like infinity or boundlessness, offering an intuitive grasp for learners.

Inequality Conversion

Converting from interval notation to inequality form is a fundamental skill in algebra that aids in understanding and solving mathematical expressions. This process involves interpreting the symbols in interval notation to form a standard inequality statement.Here’s how to do it:

• Note the endpoints in the interval notation.
• Use the less than \( < \) or greater than \( > \) symbols for open intervals and less than or equal to \( \leq \) or greater than or equal to \( \geq \) symbols for closed intervals.

Let's consider the interval \([-5, \infty)\). The bracket at \(-5\) suggests that \(-5\) is included, leading to the inequality \( x \geq -5 \). The use of \( \infty \) implies no restriction on how large \( x \) can be, perfectly embodied in this inequality. Teaching this conversion helps students transition effortlessly between different mathematical languages.